The error model for the MS gate finds its sources in the residual spin-phonon entanglement with the different levels of oscillation of the ion chain, as well as fluctuations on the intensity and phase of the laser beam and the global magnetic fields - see [1]} for a recent comprehensive analysis. An effective error model was derived in [2]} from microscopic sources and consists of the stochastic application of Pauli-type errors. The details of this error model can be found in Appendix .
